# A New Method for Dynamic Multi-Objective Optimization Based on Segment and Cloud Prediction

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Description of Dynamic Multi-Objective Problem

**Definition**

**1.**

**.**For some time t, if individual p, it can as $p\prec q$. If and only if $\forall \text{}i=\{1,2,\dots ,m\}$: ${f}_{i}(p,t)\le {f}_{i}(q,t)$, $\exists \text{}\mathrm{j}=\{1,2,\dots ,m\}$:${f}_{j}(p,t)<{f}_{j}(q,t)$.

**Definition**

**2.**

**.**Let $x\in \mathsf{\Omega}$, which is the decision variable. PS could be defined as follows,

**Definition**

**3.**

**.**Let $x\in \mathsf{\Omega}$, which is the decision variable. PF could be defined as follows,

## 3. Segmentation Cloud Prediction Strategy

#### 3.1. Population Segmentation

^{2}). Therefore, the proposed method of segmentation is simpler for population segmentation and has lower computation complexity.

#### 3.2. Directional Cloud Prediction Strategy

_{i}contains K

_{i}individuals. The center of sub-population Pop

_{i}could be described as follows,

**d**

_{i}(t) and the predicted error of two migrations $\mathbf{\Delta}\left(t\right)$ at time t+1 could be calculated as follows, based on the predictions of center positions of populations at time t and time t-1.

**v**

_{1}, and the individual position of prediction after change is y.

#### 3.3. Extra Angle Search

_{1}*, …, En

_{n}*). Then, the normal random vector ${v}_{2}~N\left({\mathit{e}}_{i}\right(t),En{*}^{2})$ can be generated, whose expectation is ${\mathit{e}}_{i}\left(t\right)$ and standard deviation is $En*$. The position of prediction from the angular deviation search can be calculated as follow,

_{2}, and the position of prediction after change is y*.

#### 3.4. Environmental Detection

#### 3.5. SCPS Framework

**Input:**When enough historical information cannot be collected, the proportion of the population in random initialization is $\zeta $, the proportion of directional prediction population is L

_{1}, the population size is N, and the final time of environmental change is T

_{max}, t:= 0, d(t − 1):= 0.

**Output:**PS.

**Step 1:**Randomly initialize the population.

**Step 2:**According to Equation (11), detect if the environment changes or not. If change, turn to Step 3; otherwise, turn to Step 8.

**Step 3:**If the value of d(t − 1) is 0, turn to Step 4; otherwise turn to Step 5.

**Step 4:**Randomly select $\zeta $ $\times $ N individuals to evolve, let d(t − 1) = d(t).

**Step 5:**According to Section 1, segment the population into m + 1 parts. Calculate the center of the population by Equation (4), and calculate the moving direction of population at time t. Then select L

_{1}× N individuals with the binary tournament selection model and predict the position with the cloud model.

**Step 6:**Select N$\times $(1 − L

_{1}) individuals with the binary tournament selection model and calculate the search vector of extra angle search for these individuals using Equation (7) to Equation (9).

**Step 7:**Boundary detect for new individuals.

**Step 8:**Calculate the non-dominated sort and crowding distance. Hold the first n individuals.

**Step 9:**Termination conditional judgment. If meet, turn to Step 10. Otherwise turn to Step 2.

**Step 10:**End, output the population PS.

_{1}$\times $ N and N $\times $ (1 − K

_{1}) individuals to predict, respectively. As the predicted individuals might be beyond the decision space, the Step 7 is to boundary detect new individuals, and revise the ones beyond.

## 4. Experimental Analysis

#### 4.1. Benchmark Problems

#### 4.2. Parameter Setting

_{max}= 10.

_{1}= 0.5N.

#### 4.3. Metrics

#### 4.4. Test and Analysis

^{−2}magnitude. That is, all these four algorithms have good ability to track the dynamic front.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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Function | n, τ | SCPS | MDP | PSS | MOEAD | ||||
---|---|---|---|---|---|---|---|---|---|

Mean Value | Variance | Mean Value | Variance | Mean Value | Variance | Mean Value | Variance | ||

JY1 | 5,10 | 1.75 × 10^{−2} | 3.99 × 10^{−4} | 4.87 × 10^{−2} | 1.18 × 10^{−3} | 2.03 × 10^{−2} | 4.79 × 10^{−3} | 5.16 × 10^{−2} | 2.06 × 10^{−3} |

10,10 | 7.55 × 10^{−3} | 1.31 × 10^{−4} | 1.64 × 10^{−2} | 3.86 × 10^{−4} | 3.17 × 10^{−2} | 9.98 × 10^{−3} | 3.43 × 10^{−2} | 1.16 × 10^{−3} | |

10,20 | 4.28 × 10^{−3} | 6.01 × 10^{−5} | 8.50 × 10^{−3} | 1.30 × 10^{−4} | 8.73 × 10^{−3} | 1.53 × 10^{−3} | 2.34 × 10^{−2} | 1.77 × 10^{−3} | |

JY2 | 5,10 | 4.76 × 10^{−2} | 2.15 × 10^{−4} | 6.17 × 10^{−2} | 7.78 × 10^{−4} | 5.09 × 10^{−2} | 9.68 × 10^{−4} | 7.15 × 10^{−2} | 1.27 × 10^{−3} |

10,10 | 7.39 × 10^{−3} | 5.10 × 10^{−5} | 1.66 × 10^{−2} | 3.06 × 10^{−4} | 3.09 × 10^{−2} | 1.38 × 10^{−2} | 3.30 × 10^{−2} | 6.96 × 10^{−4} | |

10,20 | 4.13 × 10^{−3} | 3.05 × 10^{−5} | 8.30 × 10^{−3} | 1.28 × 10^{−4} | 8.47 × 10^{−3} | 1.27 × 10^{−3} | 2.48 × 10^{−2} | 1.83 × 10^{−3} | |

JY3 | 5,10 | 9.17 × 10^{−3} | 4.36 × 10^{−4} | 1.75 × 10^{−2} | 3.46 × 10^{−4} | 1.52 × 10^{−1} | 1.33 × 10^{−1} | 2.12 × 10^{−1} | 1.86 × 10^{−1} |

10,10 | 9.11 × 10^{−3} | 6.18 × 10^{−4} | 1.74 × 10^{−2} | 3.07 × 10^{−4} | 2.91 × 10^{−1} | 2.38 × 10^{−1} | 1.87 × 10^{−1} | 1.78 × 10^{−1} | |

10,20 | 5.09 × 10^{−3} | 7.72 × 10^{−5} | 1.34 × 10^{−2} | 1.74 × 10^{−4} | 1.26 × 10^{−2} | 1.60 × 10^{−3} | 1.02 × 10^{−1} | 1.30 × 10^{−1} | |

JY4 | 5,10 | 4.07 × 10^{−2} | 1.05 × 10^{−3} | 8.55 × 10^{−2} | 1.57 × 10^{−3} | 3.85 × 10^{−2} | 5.97 × 10^{−3} | 5.66 × 10^{−2} | 2.03 × 10^{−3} |

10,10 | 2.52 × 10^{−2} | 5.52 × 10^{−4} | 5.36 × 10^{−2} | 5.52 × 10^{−4} | 5.25 × 10^{−2} | 7.97 × 10^{−3} | 3.72 × 10^{−2} | 1.15 × 10^{−3} | |

10,20 | 3.41 × 10^{−3} | 6.64 × 10^{−5} | 5.43 × 10^{−3} | 6.43 × 10^{−5} | 7.47 × 10^{−3} | 1.01 × 10^{−3} | 1.07 × 10^{−2} | 4.64 × 10^{−4} | |

JY5 | 5,10 | 2.42 × 10^{−3} | 5.83 × 10^{−6} | 1.13 × 10^{−2} | 2.38 × 10^{−4} | 1.82 × 10^{−2} | 9.69 × 10^{−3} | 3.48 × 10^{−2} | 6.36 × 10^{−4} |

10,10 | 2.42 × 10^{−3} | 3.29 × 10^{−6} | 1.10 × 10^{−2} | 2.06 × 10^{−4} | 2.11 × 10^{−2} | 1.31 × 10^{−2} | 2.46 × 10^{−2} | 1.03 × 10^{−3} | |

10,20 | 2.41 × 10^{−3} | 7.04 × 10^{−6} | 6.03 × 10^{−3} | 5.48 × 10^{−5} | 7.41 × 10^{−3} | 2.17 × 10^{−3} | 2.06 × 10^{−2} | 1.52 × 10^{−4} | |

JY6 | 5,10 | 2.11 | 6.81 × 10^{−2} | 5.90 | 1.65 × 10^{−1} | 2.97 | 1.96 × 10^{−1} | 3.35 | 1.18 × 10^{−1} |

10,10 | 1.61 | 6.26 × 10^{−2} | 3.34 | 6.99 × 10^{−2} | 3.23 | 3.76 × 10^{−1} | 2.04 | 1.25 × 10^{−1} | |

10,20 | 3.88 × 10^{−1} | 1.34 × 10^{−2} | 1.18 | 3.45 × 10^{−2} | 7.97 × 10^{−1} | 8.59 × 10^{−2} | 5.36 × 10^{−1} | 1.49 × 10^{−2} | |

JY7 | 5,10 | 1.42 | 1.26 × 10^{−1} | 16.6 | 1.68 | 89.8 | 64.7 | 20.51 | 3.76 |

10,10 | 2.06 × 10^{−1} | 9.54 × 10^{−2} | 4.33 × 10^{−1} | 5.27 × 10^{−2} | 83.9 | 67.6 | 8.38 × 10^{−1} | 5.71 × 10^{−1} | |

10,20 | 8.55 × 10^{−2} | 4.93 × 10^{−3} | 2.40 × 10^{−1} | 2.10 × 10^{−2} | 5.80 | 7.09 | 3.35 × 10^{−1} | 8.29 × 10^{−2} | |

JY8 | 5,10 | 3.34 × 10^{−3} | 4.25 × 10^{−5} | 1.12 × 10^{−2} | 2.12 × 10^{−4} | 1.93 × 10^{−2} | 1.38 × 10^{−2} | 6.31 × 10^{−2} | 3.93 × 10^{−3} |

10,10 | 3.17 × 10^{−3} | 3.45 × 10^{−5} | 1.13 × 10^{−2} | 2.23 × 10^{−4} | 2.86 × 10^{−2} | 1.44 × 10^{−2} | 6.34 × 10^{−2} | 1.92 × 10^{−2} | |

10,20 | 2.77 × 10^{−3} | 2.06 × 10^{−5} | 6.72 × 10^{−3} | 1.19 × 10^{−4} | 7.57 × 10^{−3} | 1.93 × 10^{−3} | 4.21 × 10^{−2} | 1.47 × 10^{−3} |

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**MDPI and ACS Style**

Ni, P.; Gao, J.; Song, Y.; Quan, W.; Xing, Q.
A New Method for Dynamic Multi-Objective Optimization Based on Segment and Cloud Prediction. *Symmetry* **2020**, *12*, 465.
https://doi.org/10.3390/sym12030465

**AMA Style**

Ni P, Gao J, Song Y, Quan W, Xing Q.
A New Method for Dynamic Multi-Objective Optimization Based on Segment and Cloud Prediction. *Symmetry*. 2020; 12(3):465.
https://doi.org/10.3390/sym12030465

**Chicago/Turabian Style**

Ni, Peng, Jiale Gao, Yafei Song, Wen Quan, and Qinghua Xing.
2020. "A New Method for Dynamic Multi-Objective Optimization Based on Segment and Cloud Prediction" *Symmetry* 12, no. 3: 465.
https://doi.org/10.3390/sym12030465